Towards Algorithmic Cut-Introduction

  • Stefan Hetzl
  • Alexander Leitsch
  • Daniel Weller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7180)


We describe a method for abbreviating an analytic proof in classical first-order logic by the introduction of a lemma. Our algorithm is based on first computing a compressed representation of the terms present in the analytic proof and then a cut-formula that realizes such a compression. This method can be applied to the output of automated theorem provers, which typically produce analytic proofs.


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  1. 1.
    Baaz, M., Zach, R.: Algorithmic Structuring of Cut-free Proofs. In: Martini, S., Börger, E., Kleine Büning, H., Jäger, G., Richter, M.M. (eds.) CSL 1992. LNCS, vol. 702, pp. 29–42. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  2. 2.
    Finger, M., Gabbay, D.: Equal Rights for the Cut: Computable Non-analytic Cuts in Cut-based Proofs. Logic Journal of the IGPL 15(5–6), 553–575 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Gentzen, G.: Untersuchungen über das logische Schließen. Mathematische Zeitschrift 39, 176–210, 405–431 (1934–1935)Google Scholar
  4. 4.
    Hetzl, S.: Describing proofs by short tautologies. Annals of Pure and Applied Logic 159(1–2), 129–145 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Hetzl, S.: Applying Tree Languages in Proof Theory. In: Dediu, A.-H., Martín-Vide, C. (eds.) LATA 2012. LNCS, vol. 7183, pp. 301–312. Springer, Heidelberg (2012)Google Scholar
  6. 6.
    Hetzl, S., Leitsch, A., Weller, D.: Towards Algorithmic Cut-Introduction. technical report,
  7. 7.
    Hetzl, S., Leitsch, A., Weller, D., Woltzenlogel Paleo, B.: Herbrand Sequent Extraction. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds.) AISC/Calculemus/MKM 2008. LNCS (LNAI), vol. 5144, pp. 462–477. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Miller, D., Nigam, V.: Incorporating Tables into Proofs. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 466–480. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Orevkov, V.P.: Lower bounds for increasing complexity of derivations after cut elimination. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta 88, 137–161 (1979)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Pudlák, P.: The Lengths of Proofs. In: Buss, S. (ed.) Handbook of Proof Theory, pp. 547–637. Elsevier (1998)Google Scholar
  11. 11.
    Statman, R.: Lower bounds on Herbrand’s theorem. Proceedings of the American Mathematical Society 75, 104–107 (1979)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press (2000)Google Scholar
  13. 13.
    Vyskočil, J., Stanovský, D., Urban, J.: Automated Proof Compression by Invention of New Definitions. In: Clarke, E.M., Voronkov, A. (eds.) LPAR-16 2010. LNCS, vol. 6355, pp. 447–462. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Woltzenlogel Paleo, B.: Atomic Cut Introduction by Resolution: Proof Structuring and Compression. In: Clarke, E.M., Voronkov, A. (eds.) LPAR-16 2010. LNCS, vol. 6355, pp. 463–480. Springer, Heidelberg (2010)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Stefan Hetzl
    • 1
  • Alexander Leitsch
    • 2
  • Daniel Weller
    • 1
  1. 1.Institut für Diskrete Mathematik und GeometrieTechnische Universität WienAustria
  2. 2.Institut für ComputersprachenTechnische Universität WienAustria

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